Question

$X,Y$ are mid-points of opposite sides $AB$ and $DC$ of a parallelogram $ABCD.AY$ and $DX$ are joined intersecting in P; $CX$ and $BY$ are joined intersecting in Q. What is $PXQY$?

• A. Rectangle
• B. Rhombus
• C. Parallelogram
• D. Square

Answer 1

Answer: (C)

Parallelogram

$ABCD$ is the given parallelogram

$AB=CD$ ..... (Opposite sides of parallelogram are equal)

So, $\frac{1}{2}AB=\frac{1}{2}CD$

$AX=CY$ ...... (as $X$ is the mid point of $AB$ and $Y$ is the mid point of $CD$)

$AX||CY$ ......... (We know AB||CD, as $ABCD$ is the parallelogram)

Quadrilateral $AXCY$ is a prallelogram, as a pair of opposite sides is equal and parallel.

So, $XC=YA$ and $XC\parallel YA$

$XQ\parallel YP$ ...... $\left(i\right)$ (As $XQ$ is part of line $XC$ and $YP$ is part of line $YA$)

We have $ABCD$ as the given parallelogram

Now, $AB=CD$ ....... (Opposite sides of parallelogram are equal)

So, $\frac{1}{2}AB=\frac{1}{2}CD$

$DY=XB$ ....... (As $X$ is the midpoint of $AB$ and $Y$ is the mid point of $CD$)

$DY\parallel XB$ ..... (We know $AB\parallel CD$, as $ABCD$ is the parallelogram)

Quadrilateral $DXBY$ is a prallelogram, as a pair of opposite sides is equal and parallel.

So, $DX=YB$ and $DX||YB$

$PX||YQ$ ...... $\left(ii\right)$ (As $XP$ is part of line $XD$ and $YQ$ is part of line $YB$)

From $\left(i\right)$ and $\left(ii\right)$ we get,

$PXQY$ is also a parallelogram.